Zero correlations and averaged fields of orthonormal Gaussian functions

Abstract

We consider the family of point processes $\{{\mathcal{Z}}_{f_n}{\}}_{n=0}^{\infty }$ of zeros of Gaussian random functions $\{f_n(z,\overline{z}){\}}_{n=0}^{\infty }$, arising from the Gaussian Entire Function \[ f_{0}(z):=\sum_{k=0}^{\infty} ζ_{k} \frac{z^{k}}{\sqrt{k!}}, \quad ζ_{k} \sim N_{\mathbb{C}}(0,1)\text{ i.i.d.} \] by iteration of the Landau raising operator, and orthonormal at each point in expectation in the sense that \[ \mathbb{E}\left[ e^{-\left\vert z\right\vert^{2}}f_{n}(z,\overline{z})\overline{f_{n^{\prime }}(z,\overline{z})}\right] ={δ}_{nn'}. \] We first show that the normalized pair correlations $g_{n,n+k}(z,w)$ of the pairs $({\mathcal{Z}}_{f_n},{\mathcal{Z}}_{f_{n+k}})$ exhibit \emph{a pattern reminiscent of the classical interlacing of zeros of orthogonal polynomials}: when $w\to z$, $g_{n,n+k}$ displays repulsion for $k=1$, attraction for $k=2$, and no short-range second-order correlation for $k\ge 3$. We complement this with the convergence of real-valued averaged fields on compacts $K\subset \mathbb{C}$, \[ \lim_{N \to \infty} \frac{1}{N}\sum_{n=0}^{N-1}\left\vert f_{n}(z,\overline{z})e^{-\frac{\left\vert z\right\vert^{2}}{2}} \right\vert^{2} \rightarrow 1 \quad \text{ almost surely in $C(K)$}, \] and a functional central limit theorem for the corresponding scaled fluctuations, which converge to the Gaussian process \[\mathcal{G}(z) = \frac{1}{\sqrtπ} \int_{\mathbb{C}} \mathbf{1}_{B(z,1)}(u) dW_{\mathbb{R}}(u), \] where $W_{\mathbb{R}}$ denotes real white noise on $\mathbb{C}$ and $B(z,1)$ is the unit disk centered at $z$. The results are motivated by problems in signal processing. Due to an identification with white noise spectrograms, they confirm conjectures of Flandrin and Bayram-Baraniuk and provide a rationale for the efficiency of high resolution time-frequency algorithms, namely \emph{ConceFT}, by Daubechies, Wang and Wu.

48 pages, 2 figures